The rational cuspidal divisor class group of $X_0(N)$

For any positive integer $N$, we completely determine the structure of the rational cuspidal divisor class group of $X_0(N)$, which is conjecturally equal to the rational torsion subgroup of $J_0(N)$. More specifically, for a given prime $\mathrm{\ell }$, we construct a rational cuspidal divisor $Z_{\mathrm{\ell }}(d)$ for any non-trivial divisor $d$ of $N$. Also, we compute the order of the linear equivalence class of the divisor $Z_{\mathrm{\ell }}(d)$ and show that the $\mathrm{\ell }$-primary subgroup of the rational cuspidal divisor class group of $X_0(N)$ is isomorphic to the direct sum of the cyclic subgroups generated by the linear equivalence classes of the divisors $Z_{\mathrm{\ell }}(d)$.